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Additional quasi-stationary distributions for semi-Markov processes

Published online by Cambridge University Press:  14 July 2016

David C. Flaspohler  and
Paul T. Holmes
Paul T. Holmes*
Affiliation:
Clemson University
*
*Now at Xavier University, Cincinnati, Ohio.
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Abstract

Consider a semi-Markov process X(t) defined on a subset of the non-negative integers with zero as an absorbing state and the non-zero states forming an irreducible class with exit to zero being possible. Conditions are given for the existence of the limits: where Xj(t) is the amount of time prior to time t spent in state j.

The limits (which are independent of the initial state) are evaluated when the sufficient conditions are satisfied.

Keywords

QUASI-STATIONARY DISTRIBUTIONSSEMI-MARKOV PROCESSESLIMITING CONDITIONAL DISTRIBUTIONSMARKOV RENEWAL PROCESSESABSORBING PROCESSSTATIONARY CONDITIONAL DISTRIBUTIONDOUBLY LIMITING CONDITIONAL DISTRIBUTIONLIMITING CONDITIONAL MEAN RATIO
Type
Short Communications
Information
Journal of Applied Probability , Volume 9 , Issue 3 , June 1972 , pp. 671 - 676
Copyright
Copyright © Applied Probability Trust 1972 

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References

Cheong, C. K. (1968) Ergodic and ratio limit theorems for a-recurrent semi-Markov processes. Z. Wahrscheinlichkeitsth. 9, 270–286.Google Scholar
Cheong, C. K. (1970) Quasi-stationary distributions in semi-Markov processes. J. Appl. Prob. 7, 388–399. Correction J. Appl. Prob. 7, 788.Google Scholar
Darroch, J. N. and Seneta, E. (1965) On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Prob. 2, 88–100.Google Scholar
Darroch, J. N. and Seneta, E. (1967) On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196.Google Scholar
Pyke, R. and Schaufele, R. (1964) Limit theorems for Markov renewal processes. Ann. Math. Statist. 35, 1746–1764.Google Scholar
Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.Google Scholar